On General Integral Operator of Analytic Functions

Abstract

Let $E_{\beta }$ be the integral operator defined by $E_{\beta }\left(z\right){\mathrm{\hspace{0.17em}=}\left[\beta {\int }_0^z{t}^{\beta -1}{\left(f_1^{\prime }\left(t\right)\right)}^{{\alpha }_1}{\left(f_1\left(t\right)/t\right)}^{{\gamma }_1}{P}_1^{{\zeta }_1\left(t\right)}\cdots {\left(f_n^{\prime }\left(t\right)\right)}^{{\alpha }_n}{\left(f_n\left(t\right)/t\right)}^{{\gamma }_n}{P}_n^{{\zeta }_n\left(t\right)}dt\right]}^{1/\beta }$, where each of the functions $f_i$ and $P_i$ is, respectively, analytic functions and functions with positive real part defined in the open unit disk for all $i=1,\dots ,n$. The object of this paper is to obtain several univalence conditions for this integral operator. Our main results contain some interesting corollaries as special cases.